I have a set of edges as follows:
edges = {{1, 5}, {1, 72}, {5, 72}, {1, 7}, {1, 59}, {7, 59}, {1, 8}, {1,
45}, {8, 45}, {1, 10}, {1, 73}, {10, 73}, {1, 12}, {1, 18}, {12,
18}, {1, 13}, {1, 15}, {13, 15}, {1, 15}, {1, 29}, {15, 29}, {1,
19}, {1, 22}, {19, 22}, {1, 28}, {1, 35}, {28, 35}, {1, 37}, {1,
38}, {37, 38}, {1, 43}, {1, 52}, {43, 52}, {1, 48}, {1, 67}, {48,
67}, {1, 59}, {1, 71}, {59, 71}, {2, 3}, {2, 51}, {3, 51}, {2,
7}, {2, 36}, {7, 36}, {2, 11}, {2, 67}, {11, 67}, {2, 13}, {2,
56}, {13, 56}, {2, 17}, {2, 64}, {17, 64}, {2, 18}, {2, 35}, {18,
35}, {2, 20}, {2, 25}, {20, 25}, {2, 20}, {2, 37}, {20, 37}, {2,
20}, {2, 74}, {20, 74}, {2, 24}, {2, 35}, {24, 35}, {2, 26}, {2,
49}, {26, 49}, {2, 30}, {2, 42}, {30, 42}, {2, 32}, {2, 67}, {32,
67}, {2, 37}, {2, 49}, {37, 49}, {2, 40}, {2, 47}, {40, 47}, {3,
4}, {3, 74}, {4, 74}, {3, 13}, {3, 51}, {13, 51}, {3, 14}, {3,
70}, {14, 70}, {3, 16}, {3, 45}, {16, 45}, {3, 18}, {3, 45}, {18,
45}, {3, 30}, {3, 63}, {30, 63}, {3, 31}, {3, 65}, {31, 65}, {3,
44}, {3, 45}, {44, 45}, {4, 5}, {4, 29}, {5, 29}, {4, 6}, {4,
9}, {6, 9}, {4, 6}, {4, 67}, {6, 67}, {4, 10}, {4, 74}, {10,
74}, {4, 12}, {4, 60}, {12, 60}, {4, 13}, {4, 69}, {13, 69}, {4,
16}, {4, 24}, {16, 24}, {4, 18}, {4, 55}, {18, 55}, {4, 19}, {4,
65}, {19, 65}, {4, 28}, {4, 39}, {28, 39}, {4, 32}, {4, 55}, {32,
55}, {4, 32}, {4, 64}, {32, 64}, {4, 33}, {4, 62}, {33, 62}, {4,
43}, {4, 50}, {43, 50}, {4, 51}, {4, 70}, {51, 70}, {4, 56}, {4,
71}, {56, 71}, {5, 9}, {5, 12}, {9, 12}, {5, 9}, {5, 61}, {9,
61}, {5, 10}, {5, 48}, {10, 48}, {5, 20}, {5, 61}, {20, 61}, {5,
21}, {5, 67}, {21, 67}, {5, 24}, {5, 75}, {24, 75}, {5, 32}, {5,
45}, {32, 45}, {5, 41}, {5, 45}, {41, 45}, {5, 41}, {5, 53}, {41,
53}, {5, 44}, {5, 51}, {44, 51}, {5, 48}, {5, 53}, {48, 53}, {5,
49}, {5, 64}, {49, 64}, {5, 53}, {5, 61}, {53, 61}, {6, 7}, {6,
69}, {7, 69}, {6, 10}, {6, 62}, {10, 62}, {6, 12}, {6, 56}, {12,
56}, {6, 12}, {6, 58}, {12, 58}, {6, 12}, {6, 75}, {12, 75}, {6,
17}, {6, 69}, {17, 69}, {6, 18}, {6, 32}, {18, 32}, {6, 18}, {6,
44}, {18, 44}, {6, 20}, {6, 51}, {20, 51}, {6, 29}, {6, 52}, {29,
52}};
Now I wish to chose 3
vertices v1
v2
and v3
such that there exists at least one edge between any two of these vertices and the number of edges among these 3
vertices is maximum. I can get the frequencies as follows:
frequencies = ReverseSortBy[Tally@edges, Last];
which gives me
{{{6, 12}, 3}, {{5, 61}, 3}, {{5, 53}, 3}, {{3, 45}, 3}, {{2, 20},
3}, {{6, 69}, 2}, {{6, 18}, 2}, {{5, 48}, 2}, {{5, 45},
2}, {{5, 41}, 2}, {{5, 9}, 2}, {{4, 74}, 2}, {{4, 55}, 2}, {{4, 32},
2}, {{4, 6}, 2}, {{3, 51}, 2}, {{2, 67}, 2}, {{2, 49},
2}, {{2, 37}, 2}, {{2, 35}, 2}, {{1, 59}, 2}, {{1, 15},
2}, {{59, 71}, 1}, {{56, 71}, 1}, {{53, 61}, 1}, {{51, 70},
1}, {{49, 64}, 1}, {{48, 67}, 1}, {{48, 53}, 1}, {{44, 51},
1}, {{44, 45}, 1}, {{43, 52}, 1}, {{43, 50}, 1}, {{41, 53},
1}, {{41, 45}, 1}, {{40, 47}, 1}, {{37, 49}, 1}, {{37, 38},
1}, {{33, 62}, 1}, {{32, 67}, 1}, {{32, 64}, 1}, {{32, 55},
1}, {{32, 45}, 1}, {{31, 65}, 1}, {{30, 63}, 1}, {{30, 42},
1}, {{29, 52}, 1}, {{28, 39}, 1}, {{28, 35}, 1}, {{26, 49},
1}, {{24, 75}, 1}, {{24, 35}, 1}, {{21, 67}, 1}, {{20, 74},
1}, {{20, 61}, 1}, {{20, 51}, 1}, {{20, 37}, 1}, {{20, 25},
1}, {{19, 65}, 1}, {{19, 22}, 1}, {{18, 55}, 1}, {{18, 45},
1}, {{18, 44}, 1}, {{18, 35}, 1}, {{18, 32}, 1}, {{17, 69},
1}, {{17, 64}, 1}, {{16, 45}, 1}, {{16, 24}, 1}, {{15, 29},
1}, {{14, 70}, 1}, {{13, 69}, 1}, {{13, 56}, 1}, {{13, 51},
1}, {{13, 15}, 1}, {{12, 75}, 1}, {{12, 60}, 1}, {{12, 58},
1}, {{12, 56}, 1}, {{12, 18}, 1}, {{11, 67}, 1}, {{10, 74},
1}, {{10, 73}, 1}, {{10, 62}, 1}, {{10, 48}, 1}, {{9, 61},
1}, {{9, 12}, 1}, {{8, 45}, 1}, {{7, 69}, 1}, {{7, 59},
1}, {{7, 36}, 1}, {{6, 75}, 1}, {{6, 67}, 1}, {{6, 62},
1}, {{6, 58}, 1}, {{6, 56}, 1}, {{6, 52}, 1}, {{6, 51},
1}, {{6, 44}, 1}, {{6, 32}, 1}, {{6, 29}, 1}, {{6, 20},
1}, {{6, 17}, 1}, {{6, 10}, 1}, {{6, 9}, 1}, {{6, 7}, 1}, {{5, 75},
1}, {{5, 72}, 1}, {{5, 67}, 1}, {{5, 64}, 1}, {{5, 51},
1}, {{5, 49}, 1}, {{5, 44}, 1}, {{5, 32}, 1}, {{5, 29},
1}, {{5, 24}, 1}, {{5, 21}, 1}, {{5, 20}, 1}, {{5, 12},
1}, {{5, 10}, 1}, {{4, 71}, 1}, {{4, 70}, 1}, {{4, 69},
1}, {{4, 67}, 1}, {{4, 65}, 1}, {{4, 64}, 1}, {{4, 62},
1}, {{4, 60}, 1}, {{4, 56}, 1}, {{4, 51}, 1}, {{4, 50},
1}, {{4, 43}, 1}, {{4, 39}, 1}, {{4, 33}, 1}, {{4, 29},
1}, {{4, 28}, 1}, {{4, 24}, 1}, {{4, 19}, 1}, {{4, 18},
1}, {{4, 16}, 1}, {{4, 13}, 1}, {{4, 12}, 1}, {{4, 10}, 1}, {{4, 9},
1}, {{4, 5}, 1}, {{3, 74}, 1}, {{3, 70}, 1}, {{3, 65},
1}, {{3, 63}, 1}, {{3, 44}, 1}, {{3, 31}, 1}, {{3, 30},
1}, {{3, 18}, 1}, {{3, 16}, 1}, {{3, 14}, 1}, {{3, 13}, 1}, {{3, 4},
1}, {{2, 74}, 1}, {{2, 64}, 1}, {{2, 56}, 1}, {{2, 51},
1}, {{2, 47}, 1}, {{2, 42}, 1}, {{2, 40}, 1}, {{2, 36},
1}, {{2, 32}, 1}, {{2, 30}, 1}, {{2, 26}, 1}, {{2, 25},
1}, {{2, 24}, 1}, {{2, 18}, 1}, {{2, 17}, 1}, {{2, 13},
1}, {{2, 11}, 1}, {{2, 7}, 1}, {{2, 3}, 1}, {{1, 73}, 1}, {{1, 72},
1}, {{1, 71}, 1}, {{1, 67}, 1}, {{1, 52}, 1}, {{1, 48},
1}, {{1, 45}, 1}, {{1, 43}, 1}, {{1, 38}, 1}, {{1, 37},
1}, {{1, 35}, 1}, {{1, 29}, 1}, {{1, 28}, 1}, {{1, 22},
1}, {{1, 19}, 1}, {{1, 18}, 1}, {{1, 13}, 1}, {{1, 12},
1}, {{1, 10}, 1}, {{1, 8}, 1}, {{1, 7}, 1}, {{1, 5}, 1}}
By observation {6, 12}
has 3
edges and the next sorted occurrence of 6 or 12
is {6, 69}
which has 2
edges, however, {12, 69}
has no edge. For {5, 61}
, the next occurrence is {5, 53}
and both have 3
edges each and also there is an edge connecting {53, 61}
. How can I choose these three variables {5, 53, 61}
?
"IGraphM`"
provides the function that can do that. $\endgroup$